Invariant Solutions, Conservation Laws, and Jacobi Elliptic Waves for a Fifth-Order Nonlinear Evolution Equation

Abstract

A detailed structural analysis of a fifth-order nonlinear evolution equation is presented in this work. The study begins with the determination of classical Lie point symmetries, leading to the construction of an optimal system of one-dimensional subalgebras and the derivation of corresponding similarity reductions. These reductions transform the governing partial differential equation into a nonlinear ordinary differential equation that characterizes invariant solution profiles. The nonclassical symmetry approach is subsequently employed to uncover additional invariant structures that are not accessible through the classical framework, yielding complex exponential and traveling-wave type solutions. To further explore the intrinsic properties of the equation, conservation laws are systematically derived via Ibragimov’s theorem. The existence of multiple conserved vectors highlights the internal structural consistency of the model and supports its integrable nature. In addition, explicit analytical solutions are obtained using the modified Jacobi elliptic expansion method. The resulting families of solutions include periodic and solitary-type wave forms expressed in terms of hyperbolic and elliptic functions. The combined application of symmetry analysis, conservation law construction, and elliptic function expansion provides a comprehensive understanding of the invariant structure and solution space of the considered fifth-order nonlinear model.